Necessary conditions for the existence of exponential-polynomial expansions for solutions of certain differential equations

We consider ordinary differential equations (ODE) of the form $u^{\mathrm{\prime }\mathrm{\prime }}u-(u^{\mathrm{\prime }}{)}^2=e^{-x}P(u)-1$, where $P$ is a polynomial. For $P=u^k$, $k=3,4,6$ this ODE is equivalent to certain degenerate Painlevé III equations. We study whether families of solutions of these ODEs have asymptotic expansions of the form $u(x)={\sum }_{k=0}^{\mathrm{\infty }}{p}_k(x+c)e^{-kx}$ for $Rex\to +\mathrm{\infty }$, where $c\in \mathbb{C}$ is an arbitrary constant parameterizing the solution family, $p_k$ are polynomials, with $p_0(x)=x$. We find necessary conditions on $P$ for such expansions to exist. Numerical experiments suggest that these conditions are also sufficient, and the expansions are not only formal, but actually provide a series representation of the solutions. Numerical evidence also suggests a conjecture on the nonnegativity of coefficients of the $p_k$.